Truth-Value Gaps and Meaning
Sentences exhibiting truth-value gaps would appear to pose a significant problem for truth-conditional semantics. Such sentences evidently have meaning, yet they are neither true nor false. In this respect they resemble non-indicative sentences such as imperatives. But imperatives can be handled by adopting a parallel concept like obedience conditions and proceeding in the usual way.  How do we deal with sentences like “The king of France is bald” or “Colorless green ideas sleep furiously” or “All my parakeets are asleep” (said when I have no parakeets). These sentences are as meaningful as any, yet they lack truth-value.  And there are infinitely many of them, as many as there are sentences with truth-value. Should we conclude that Tarski-style semantics for them is impossible? They don’t even have falsity conditions, so how could they submit to a recursive definition of the kind Tarski showed how to provide? We understand such sentences—our linguistic mastery encompasses them—and we also appreciate that they lack truth-value, so how can truth-based semantics apply to them?
It is an interesting fact that there is no simple predicate capturing the condition of being neither true nor false, so theorists adopt the makeshift “gappy”, or we could stipulate a use for “vacuous” applied to whole sentences (as in “vacuous names”). For convenience I will abbreviate “neither true nor false” to “NTF”, so that I can say that a sentence s is NTF if and only if p, where p is some sentence in the meta-language yet to be specified. The question is what that sentence will be. For truth we simply repeat the sentence of the object language (or a translation of it), for falsehood we prefix the sentence on the right with negation—what do we do for “NTF”? What we need is a necessary and sufficient condition for the semantic predicate “NTF” to apply. It seems fairly obvious what this should be: s is NTF if and only if it is not the case that either p or not-p, where p is (or translates) s. For example, “The king of France is bald” is NTF if and only if it is not the case that the king of France is either bald or that he is not bald. That is, the law of excluded middle doesn’t apply to the sentence in question. If there is no king of France, he can’t be either bald or not bald, so a sentence affirming that he is bald is neither true nor false.  Notice that the condition on the right hand side is not meta-linguistic, so it resembles the usual disquotational conditionals made famous by Tarski. We could say that “snow is white” is made true by the fact that snow is white, “snow is black” is made false by the fact that snow is not black, and “The king of France is bald” is made neither true nor false by the fact that he is neither bald nor not bald. Similarly, it is not a fact that colorless green ideas sleep furiously or that they don’t, so the sentence stating this is neither true not false. When a speaker understands such a sentence she knows that the facts don’t give it a determinate truth-value, and her understanding is displayed by the biconditional enunciated. We have the usual mention-use pattern of classical truth theories, but the right hand side doesn’t just repeat the left—it provides a more complex condition. The same is true for falsity, because there we have to add negation. Not all semantic biconditionals are “homophonic”.
Employing this basic format, we can provide recursive clauses in the usual manner. Thus “p and q” is NTF if and only if both p and q are NTF; “p or q” is NTF if and only if either p or q is NTF ; “not-p” is NTF if and only if p is NTF. To deal with quantified sentences we introduce the notion of a “true of” (satisfaction) gap: the predicate is neither true nor false of a putative object (such as a French monarch). The reference of the description is neither bald nor not bald, since there is no such reference. Compare “Vulcan revolves”: the putative planet Vulcan neither revolves nor fails to revolve, so it doesn’t satisfy “revolve” or dissatisfy it. Thus we can apply the standard Tarskian apparatus to the concept of a truth-value gap, mutatis mutandis. We can therefore provide a recursive disquotational definition of the predicate “NTF”. We could call the form of this definition “Convention NTF” and require that for any sentence of the object language such a meta-language sentence be derivable. Thus we have Convention T for truth, Convention F for falsity, and Convention NTF for neither truth nor falsity: the first simply repeats the sentence, the second introduces negation, while the third deploys negation plus disjunction. For “snow is white” to be true is for snow to be white, for “snow is white” to be false is for snow not to be white, and for “snow is white” to be neither true nor false is for snow neither to be white nor not white. Thus we bring sentences exhibiting truth-value gaps within the fold of Tarski-style theories—not by subsuming them under the concept of truth but by extending the apparatus beyond that concept. We might call this generalized truth-theoretic semantics.
It is a question whether every sentence has an NTF condition, not just those that are actually neither true nor false. Do we, in understanding “snow is white”, grasp under what conditions it would be neither true nor false, as we grasp its truth conditions and its falsity conditions? We grasp this for sentences that are NTF because we recognize their “gappy” status, but do we also grasp it when we know that there is nothing gappy going on? I rather think we do: for we grasp what it would be for them to be NTF. I know that “snow is white” would have a truth-value gap if snow were neither white nor not white—though I also know that it is actually one way or the other. If you ask me under what conditions the sentence would be NTF, I can tell you—if there’s no fact of the matter about the color of snow (as there is no fact of the matter about the color of Hamlet’s hair). So NTF conditions are pervasive in the understanding of language: they are part of what every speaker (tacitly) knows. Every sentence has conditions under which it would be NTF, say by the subject-term lacking a reference (and this is always an epistemic possibility ), and this is something speakers grasp at some level. So a semantics for the predicate “neither true nor false” is applicable everywhere. We all grasp something of the form: “snow is white” would be neither true nor false if and only if snow was neither white nor not white (possibly by not existing). NTF conditions are as much grasped as truth conditions and falsity conditions.
 I am going to assume the existence of meaningful sentences with truth-value gaps without arguing for it. My question is what happens to semantics if we accept such sentences. I will also not discuss all the possible examples that have been offered: vague sentences, future contingents, empty names and demonstratives, ethical sentences, etc. What I propose will carry over to these cases.
 It is worth noting that empty descriptions can also occur in imperative sentences such as “Kill the king of France!”, so we have obedience-value gaps as well as truth-value gaps. Then too we have “Bring me some colorless green ideas!”
 Actually this clause is too simple given the case in which p is true and q is NTF, since this will make the disjunction come out true. We could append “unless p is true” to cover this case, but for simplicity I will stick with the condition as stated in the text.
 In the extreme case in which we are brains in vats and our noun phrases are empty of reference, truth-value gaps will be ubiquitous, and hence the correct semantics for our language will be largely a NTF semantics.